Stability and bifurcation analysis on a Logistic model with discrete and distributed delays.
ABSTRACT In this paper, a Logistic model with discrete and distributed delays is investigated. We first consider the stability of the positive equilibrium and the existence of local Hopf bifurcations. In succession, using the normal form theory and center manifold argument, we derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions. A numerical simulation for supporting the theoretical analysis is also given.

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Page 1
Stability and bifurcation analysis on a Logistic model
with discrete and distributed delaysq
Yongli Songa,*, Yahong Pengb
aDepartment of Applied Mathematics, Tongji University, 1239 Siping Road, Shanghai 200092, PR China
bDepartment of Applied Mathematics, Donghua University, Shanghai 200051, PR China
Abstract
In this paper, a Logistic model with discrete and distributed delays is investigated. We first consider the stability of the
positive equilibrium and the existence of local Hopf bifurcations. In succession, using the normal form theory and center
manifold argument, we derive the explicit formulas which determine the stability, direction and other properties of bifur
cating periodic solutions. A numerical simulation for supporting the theoretical analysis is also given.
? 2006 Elsevier Inc. All rights reserved.
Keywords: Logistic model; Time delays; Stability; Hopf bifurcations; Periodic solutions
1. Introduction
The wellknown Logistic differential equation with discrete delay
?
and the distributed delay Logistic equation
Zt
have been widely studied (see for example [1–6] and references therein). In addition, Gopalsamy [7] also stud
ied a Logistic model with two delays
_ xðtÞ ¼ rxðtÞ 1 ?xðt ? sÞ
K
?
ð1:1Þ
_ xðtÞ ¼ rxðtÞ 1 ?xðtÞ
K
?1
Q
?1
fðt ? sÞxðsÞ ds
??
ð1:2Þ
_ xðtÞ ¼ rxðtÞð1 ? a1xðt ? sÞ ? a2xðt ? cÞÞ:
In the present paper, we consider the following more general delayed Logistic equation
ð1:3Þ
00963003/$  see front matter ? 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2006.03.025
qThis research is supported in part by the National Natural Science Foundation of China.
*Corresponding author.
Email address: syl.mail@163.com (Y. Song).
Applied Mathematics and Computation 181 (2006) 1745–1757
www.elsevier.com/locate/amc
Page 2
_ xðtÞ ¼ rxðtÞ 1 ? a1xðt ? sÞ ? a2
Zt
?1
fðt ? sÞxðsÞ ds
??
;
ð1:4Þ
where r,s,a1,a2> 0. The function f in (1.4) is called the delayed kernel, which is the weight given to the pop
ulation t time units ago, and we shall assume it satisfies f(t) P 0 for all t P 0 together with the normalization
condition
Z1
which ensures that the steady states of the model (1.4) are unaffected by the delay. Clearly, systems (1.1)–(1.3)
are the special cases of system (1.4) under some special delayed kernel f(t) such as when f(t) = d(t ? r), Eq.
(1.4) becomes Eq. (1.3). Following the ideal of Cushing [8], the weak kernel
fðtÞ ¼ re?rt;
and the strong kernel
0
fðtÞ dt ¼ 1;
r > 0
ð1:5Þ
fðtÞ ¼ r2te?rt;
are frequently encountered in the literature. In this paper, although we consider only Eq. (1.4) with the weak
kernel (1.5), the strong kernel case can be handled similarly. Taking the delay s as a parameter, we investigate
the effect of the delay s on the dynamics of Eq. (1.4). More specifically, we show that the stability switches and
a Hopf bifurcation occur when the delay s passes through a critical value.
This paper is organized as follows. In Section 2, we consider the stability and the local Hopf bifurcation of
the positive equilibrium. In Section 3, we first employ the normal form method and the center manifold theory
introduced by Hassard et al. [9] to analyze the direction, stability and the period of the bifurcating periodic
solution at the critical values of s, then a numerical example is given. We end with some conlusions in Section
4.
r > 0
ð1:6Þ
2. Stability of the positive equilibrium and existence of Hopf bifurcations
Clearly, Eq. (1.4) always has a positive equilibrium
x?¼
1
a1þ a2:
Now, we define a new variable
Zt
then by the linear chain trick technique, system (1.4) can be transformed into the following equivalent system
with a discrete delay
?
Obviously, system (2.1) has a unique positive equilibrium E*= (x*,y*) with y*= x*. By the linear transform
?
system (2.1) becomes
?
Then the linearized system of (2.2) at the zero steady state is
yðtÞ ¼
?1
re?rðt?sÞxðsÞ ds;
_ xðtÞ ¼ rxðtÞð1 ? a1xðt ? sÞ ? a2yðtÞÞ;
_ yðtÞ ¼ rxðtÞ ? ryðtÞ:
ð2:1Þ
x1ðtÞ ¼ xðtÞ ? x?;
x2ðtÞ ¼ yðtÞ ? y?;
_ x1ðtÞ ¼ ?ra1x?x1ðt ? sÞ ? ra2x?x2ðtÞ ? ra1x1ðtÞx1ðt ? sÞ ? ra2x1ðtÞx2ðtÞ;
_ x2ðtÞ ¼ rx1ðtÞ ? rx2ðtÞ:
ð2:2Þ
1746
Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757
Page 3
_ x1ðtÞ ¼ ?ra1x?x1ðt ? sÞ ? ra2x?x2ðtÞ;
_ x2ðtÞ ¼ rx1ðtÞ ? rx2ðtÞ:
The associated characteristic equation of system (2.3) is the following second degree exponential polynomial
equation:
?
ð2:3Þ
k2þ pk þ g þ ðsk þ qÞe?ks¼ 0;
where p = r > 0, g = rra2x*> 0, s = ra1x*> 0, q = rra1x*> 0.
In order to investigate the stability of the positive equilibrium of system (2.1), we need to study the distri
bution of roots of Eq. (2.4). Clearly, k = 0 is not a root of Eq. (2.4).
If ix(x > 0) is a root of Eq. (2.4), then
ð2:4Þ
g ? x2þ sxsinxs þ qcosxs þ pxi þ ½sxcosxs ? qsinxs?i ¼ 0:
Separating the real and imaginary parts, we have
?
which lead to
x2? g ¼ sxsinxs þ qcosxs;
px ¼ qsinxs ? sxcosxs;
ð2:5Þ
x4? ðs2þ 2g ? p2Þx2þ g2? q2¼ 0:
ð2:6Þ
Let
D ¼ ðs2þ 2g ? p2Þ2? 4ðg2? q2Þ:
It is easy to see that if at least one of the following is satisfied:
ðH1Þ
ðH2Þ
ðH3Þ
D < 0;
D > 0;
g2? q2P 0;
s2þ 2g ? p26 0;
s2þ 2g ? p2< 0;
D ¼ 0;
then Eq. (2.6) has no positive root; if
D > 0;
g2? q2> 0;
s2þ 2g ? p2> 0
or
D ¼ 0;
s2þ 2g ? p2> 0
holds, then Eq. (2.6) has two positive roots
ffiffiffi
and if
x?¼
2
p
2
s2þ 2g ? p2?
ffiffiffiffi
D
p
hi1
2;
g2? q2< 0
or
g2? q2¼ 0;
holds, then Eq. (2.6) has only one positive root x+.
Without loss of generality, we suppose that Eq. (2.6) has two positive roots ±x±. Then, from (2.5), we can
determine
ðq ? psÞx2
s2x2
s2þ 2g ? p2> 0
s?
j¼
1
x?
cos?1
?? gq
?þ q2
??
þ2jp
x?;
j ¼ 0;1;...;
ð2:7Þ
at which Eq. (2.4) has a pair of purely imaginary roots ±ix±. Denote by
kðsÞ ¼ aðsÞ þ ixðsÞ
Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757
1747
Page 4
the root of Eq. (2.4) such that
aðs?
jÞ ¼ 0;
xðs?
jÞ ¼ x?:
Substituting k(s) into (2.4) and taking the derivative with respect to s, we have
?
which, together with (2.5), leads to
?
pcosx?s?
dk
ds
??1
¼ð2k þ pÞeks
kðsk þ qÞþ
s
kðsk þ qÞ?s
k;
Redk
ds
??1
s¼s?
j
¼ Re
ð2k þ pÞeks
kðsk þ qÞ
?
?
?sx2
n
p2x2
?
s¼s?
j
þ Re
s
kðsk þ qÞ
jþ i½2x?cosx?s?
?sx2
j? 2x?sinx?s?
??
s¼s?
j
¼ Re
¼1
C
¼1
C
¼1
C
¼x2
j? 2x?sinx?s?
jþ psinx?s?
j?
?þ iqx?
j? þ qx?½2x?cosx?s?
?
þ Re
s
?sx2
j? ? s2x2
?þ iqx?
o
o
??
?½pcosx?s?
jþ psinx?s?
?
n
px?½qsinx?s?
j? sx?cosx?s?
j? þ 2x2
?¼x2
?½qcosx?s?
jþ sx?sinx?s?
j? ? s2x2
?
?þ 2x4
?? 2gx2
?? s2x2
ffiffiffiffi
?
?
?
?
C
2x2
?þ ðp2? s2? 2gÞ
¼x2
C
??
C
s2? p2þ 2g ?
D
p
þ ðp2? s2? 2gÞ
no
?
?
ffiffiffiffi
D
p
no
;
where C ¼ s2x4
sign Redk
?þ q2x2
?
?> 0. Thus, if D 5 0, we have
)
ds
ds
?
s¼sþ
j
(
¼ sign Redk
???1
s¼sþ
j
()
¼ sign
x2
þ
ffiffiffiffi
D
p
C
()
> 0
ð2:8Þ
and
sign Redk
ds
??
s¼s?
j
()
¼ sign Redk
ds
???1
s¼s?
j
()
¼ sign ?x2
?
ffiffiffiffi
D
p
C
()
< 0:
ð2:9Þ
Notice that when s = 0, Eq. (2.4) becomes
k2þ ðp þ sÞk þ g þ q ¼ 0;
two roots of which always have negative real part. In addition, note that
g2? q2¼ r2r2x2
Thus, from the previous discussions and the lemma in [10], we can obtain the following results about the
distribution of roots of Eq. (2.4).
?ða2þ a1Þða2? a1Þ;s2þ 2g ? p2¼ ?r2þ 2ra2x?r þ ðra1x?Þ2:
Lemma 2.1. Let s?
jðj ¼ 0;1;...Þ be defined as in (2.7).
(i) If at least one of the conditions (H1)–(H3) is satisfied, then all roots of Eq. (2.4) have negative real parts for
all s P 0.
(ii) If D > 0,a2> a1 and 0 < r <
a2þa1
sþ
s 2 ðsþ
(iii) If either a2< a1or a2= a1and 0 < r <
s 2 ½0;sþ
rða2þ
ffiffiffiffiffiffiffiffiffi
a2
2þa2
1
p
Þ
, then there exists a k 2 N such that when s 2 ðs?
j?1;
jÞ; j ¼ 0;1;...;k, all roots of Eq. (2.4) have negative real parts, where s?
j;s?
ffiffi
?1¼ 0, and when
jÞ; j ¼ 0;1;...;k ? 1, and s > sþ
k, Eq. (2.4) has at least one root with positive real part.
þ12
2r holds, then all roots of Eq. (2.4) have negative real parts for
p
0Þ, and Eq. (2.4) has at least one root with positive real part for s > sþ
0.
1748
Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757
Page 5
Applying Lemma 2.1, we then obtain the following theorem.
Theorem 2.2. Let s?
jðj ¼ 0;1;...Þ be defined as in (2.7).
(i) At least one of the conditions (H1)–(H3) is satisfied, then the positive equilibrium E*of system (2.1) (or the
positive equilibrium x*of Eq. (1.4)) is asymptotically stable.
rða2þ
1
a2þa1
switches from stability to instability, that is, the positive equilibrium E*of system (2.1) (or the positive equi
librium x*of Eq. (1.4)) is asymptotically stable for s 2Sk
(iii) If either a2< a1or a2= a1and 0 < r <
positive equilibrium x*of Eq. (1.4)) is asymptotically stable for s 2 ½0;sþ
(iv) If D > 0,a2> a1and 0 < r <
a2þa1
s?
a Hopf bifurcation at sþ
(ii) If D > 0,a2> a1and 0 < r <
ffiffiffiffiffiffiffiffiffi
a2
2þa2
p
Þ
, then there are k 2 N such that the stability of E*of system (2.1)
j¼0ðs?
j?1;sþ
jÞ, where s?
?1¼ 0, and unstable for
s 2Sk?1
j¼0ðsþ
j;s?
jÞ and s > sþ
k.
ffiffi
2
2r holds, then the positive equilibrium E*of system (2.1) (or the
0Þ, and unstable for s > sþ
, then system (2.1) (or Eq. (1.4)) undergoes a Hopf bifurcation at
ffiffi
p
þ1
0.
rða2þ
ffiffiffiffiffiffiffiffiffi
a2
2þa2
1
p
Þ
j. However, if either a2< a1or a2= a1and 0 < r <
j.
2
2r holds, then system (2.1) (or Eq. (1.4)) undergoes
p
þ1
3. Direction and stability of the Hopf bifurcation
In the previous section, we obtain the conditions, under which system (2.1) or Eq. (1.4) undergoes Hopf
bifurcation from the positive steady state at the critical values of s, i.e. a family of periodic solutions bifurcate
from the zero equilibrium. Using the normal form theory and center manifold reduction by Hassard et al. [9],
we are able to determine the Hopf bifurcation direction, and investigate the properties of these bifurcating
periodic solutions, for example, stability on the center manifold and period. Throughout this section, we
always assume that system (2.1) undergoes Hopf bifurcations at the critical value ~ s of s, and then ±ix is cor
responding purely imaginary roots of the characteristic equation associated with the positive steady state E*.
Letting x1= x ? x*, x2= y ? y*, ? xiðtÞ ¼ xiðstÞ, s ¼ ~ s þ l and dropping the bars for simplification of nota
tion, system (2.1) is transformed into an FDE in C = C([?1,0],R2) as
_ xðtÞ ¼ LlðxtÞ þ fðl;xtÞ;
where x(t) = (x1(t),x2(t))T2 R2, xt(h) = x(t + h) 2 C, and Ll:C ! R, f: R · C ! R are given, respectively, by
0
?ra2x?
r
?r
and
?
By the Riesz representation theorem, there exists a function g(h,l) of bounded variation for h 2 [?1,0], such
that
Z0
In fact, we can choose
?
where d is defined by
?
ð3:1Þ
LlðxtÞ ¼ ð~ s þ lÞ
??
x1tð0Þ
x2tð0Þ
??
þ ð~ s þ lÞ
?ra1x?
0
0
0
??
x1tð?1Þ
x2tð?1Þ
??
ð3:2Þ
fðs;xtÞ ¼ ð~ s þ lÞ
?ra1x?x1tð0Þx1tð?1Þ ? ra2x?x1tð0Þx2tð0Þ
0
?
:
ð3:3Þ
Ll/ ¼
?1
dgðh;0Þ/ðhÞ
for / 2 C:
ð3:4Þ
gðh;lÞ ¼ ð~ s þ lÞ
0
r
?ra2x?
?r
?
dðhÞ þ ð~ s þ lÞ
?ra1x?
0
0
0
??
dðh þ 1Þ;
ð3:5Þ
dðhÞ ¼
0;
1;
h 6¼ 0;
h ¼ 0:
Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757
1749
Page 6
For / 2 C1([?1,0],R2), define
AðlÞ/ ¼
d/ðhÞ
dh;
R0
0;
fðl;/Þ;
h 2 ½?1;0Þ;
h ¼ 0:
?1dgðl;sÞ/ðsÞ;
(
and
RðlÞ/ ¼
h 2 ½?1;0Þ;
h ¼ 0:
?
Then system (3.1) is equivalent to
_ xt¼ AðlÞxtþ RðlÞxt;
where xt(h) = x(t + h) for h 2 [?1,0].
For w 2 C1([0,1],(R2)*), define
?dwðsÞ
R0
and a bilinear inner product
ð3:6Þ
A?wðsÞ ¼
ds;
s 2 ð0;1?;
s ¼ 0
?1dgTðt;0Þwð?tÞ;
(
hwðsÞ;/ðhÞi ¼?wð0Þ/ð0Þ ?
Z0
?1
Zh
n¼0
?wðn ? hÞ dgðhÞ/ðnÞ dn;
ð3:7Þ
where g(h) = g(h,0). Then A(0) and A*are adjoint operators. By the discussion in Section 2, we know that
?ix~ s are eigenvalues of A(0). Thus, they are also eigenvalues of A*. We first need to compute the eigenvector
of A(0) and A*corresponding to ix~ s and ?ix~ s, respectively.
Suppose that qðhÞ ¼ ða;1ÞTeihx~ sis the eigenvector of A(0) corresponding to ix~ s. Then Að0ÞqðhÞ ¼ ix~ sqðhÞ.
It follows from the definition of A(0) and (3.4) and (3.5) that
?
Thus, we can easily obtain
?
On the other hand, suppose that q?ðsÞ ¼ Dð1;bÞeisx~ sis the eigenvector of A*corresponding to ?ix~ s. By the
definition of A*and (3.4) and (3.5), we have
?
which means
q?ð0Þ ¼ Dð1;bÞ ¼ D 1;?ix þ ra1x?eix~ s
r
~ s
ix þ ra1x?e?ix~ s
?r
ra2x?
ix þ r
?
qð0Þ ¼
0
0
? ?
:
qð0Þ ¼ ða;1ÞT¼
r þ ix
r
;1
?T
:
~ s
?ix þ ra1x?eix~ s
ra2x?
?r
?ix þ r
?
q?ð0ÞðÞT¼
0
0
? ?
;
??
:
In order to assure hq*(s),q(h)i = 1, we need to determine the value of D. From (3.7), we have
hq?ðsÞ;qðhÞi ¼ Dð1;?bÞða;1ÞT?
?1
Z0
Thus, we can chose D as
Z0
Zh
n¼0
Dð1;?bÞe?iðn?hÞx~ sdgðhÞða;1ÞTeinx~ sdn
?
¼ D a þ?b ?
?1ð1;?bÞheihx~ sdgðhÞða;1ÞT
?
¼ D a þ?b þ~ sraa1x?e?ix~ s
??
D ¼
1
a þ b þ~ sraa1x?eix~ s:
1750
Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757
Page 7
In the remainder of this section, we will follow the ideas and use the same notations as in [9], we first com
pute the coordinates to describe the center manifold C0at l = 0. Let xtbe the solution of Eq. (3.1) when l = 0.
Define
zðtÞ ¼ hq?;xti;
On the center manifold C0we have
W ðt;hÞ ¼ W ðzðtÞ;? zðtÞ;hÞ;
where
W ðt;hÞ ¼ xtðhÞ ? 2RefzðtÞqðhÞg:
ð3:8Þ
W ðz;? z;hÞ ¼ W20ðhÞz2
z and ? z are local coordinates for center manifold C0in the direction of q*and ? q?. Note that W is real if xtis
real. We consider only real solutions. For the solution xt2 C0of (3.1), since l = 0, we have
_ zðtÞ ¼ ix~ sz þ hq?ðhÞ;Fð0;W ðz;? z;hÞ þ 2RefzqðhÞgÞi
¼ ix~ sz þ ? q?ð0Þfð0;W ðz;? z;0Þ þ 2Refzqð0ÞgÞ ¼
We rewrite this equation as
2þ W11ðhÞz? z þ W02ðhÞ? z2
2þ ???;
defix~ sz þ ? q?ð0Þf0ðz;? zÞ:
_ zðtÞ ¼ ix~ szðtÞ þ gðz;? zÞ
with
gðz;? zÞ ¼ ? q?ð0Þf0ðz;? zÞ ¼ g20
Noticing xtðhÞ ¼ ðx1tðhÞ;x2tðhÞ;x3tðhÞÞ ¼ W ðt;hÞ þ zqðhÞ þ zqðhÞ and qðhÞ ¼ ða;1ÞTeihx~ s, we have
x1tð0Þ ¼ az þ az þ Wð1Þ
20ð?1Þz2
20ð0Þz2
z2
2þ g11z? z þ g02
? z2
2þ g21
z2? z
2þ ???ð3:9Þ
20ð0Þz2
2þ Wð1Þ
11ð0Þz? z þ Wð1Þ
02ð0Þ? z2
2þ Oðjðz;? zÞj3Þ;
02ð?1Þ? z2
x1tð?1Þ ¼ ae?ix~ sz þ aeix~ s? z þ Wð1Þ
2þ Wð1Þ
11ð?1Þz? z þ Wð1Þ
02ð0Þ? z2
2þ Oðjðz;? zÞj3Þ;
x2tð0Þ ¼ z þ? z þ Wð2Þ
2þ Wð2Þ
11ð0Þz? z þ Wð2Þ
2þ Oðjðz;? zÞj3Þ:
Thus, from (3.9), we have
gðz;? zÞ ¼ ? q?ð0Þf0ðz;? zÞ ¼ ~ sDð1;?bÞ
?ra1x?x1tð0Þx1tð?1Þ ? ra2x?x1tð0Þx2tð0Þ
0
??
¼ ?~ sDrx? a1 az þ ? az þ Wð1Þ
?
þ a2 az þ ? az þ Wð1Þ
?
¼ ?~ sDrx? 2 a1a2e?ix~ sþ a2a
?
þ a2 2aWð2Þ
20ð0Þz2
20ð?1Þz2
2þ Wð1Þ
11ð0Þz? z þ Wð1Þ
02ð0Þ? z2
02ð?1Þ? z2
2þ Oðjðz;? zÞj3Þ
???
?
ae?ix~ sz þ ? aeix~ s? z þ Wð1Þ
?
z þ? z þ Wð2Þ
?
þ a1 2aWð1Þ
?
2þ Wð1Þ
11ð?1Þz? z þ Wð1Þ
02ð0Þ? z2
02ð0Þ? z2
2þ Oðjðz;? zÞj3Þ
?
??
i
20ð0Þ
?
?
20ð0Þz2
2þ Wð1Þ
11ð0Þz? z þ Wð1Þ
2þ Oðjðz;? zÞj3Þ
?
20ð0Þz2
2þ Wð2Þ
11ð0Þz? z þ Wð2Þ
?z2
20ð?1Þ þ 2ae?ix~ sWð1Þ
2þ Oðjðz;? zÞj3Þ
2þ 2 a1jaj2Refeix~ sg þ a2Refag
h
z? z þ 2 a1? a2eix~ sþ a2? a
?
.
??? z2
2
?
11ð?1Þ þ ? aWð1Þ
11ð0Þ þ ? aeix~ sWð1Þ
?iz2? z
h
11ð0Þ þ ? aWð1Þ
20ð0Þ þ 2Wð1Þ
11ð0Þ þ Wð2Þ
20ð0Þ
2þ ???
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Comparing the coefficients with (3.9), we get
?
g11¼ ?2~ sDrx? a1jaj2Refeix~ sg þ a2Refag
g02¼ ?2~ sDrx? a1a2eix~ sþ a2a
g21¼ ?2~ sDrx? a1 2aWð1Þ
þa2 2aWð2Þ
Since there are W20(h) and W11(h) in g21, in the sequel, we shall compute them.
From (3.6) and (3.8), we have
?
where
g20¼ ?2~ sDrx? a1a2e?ix~ sþ a2a
?
n
?;
?
;
??;
11ð?1Þ þ aWð1Þ
11ð0Þ þ aWð1Þ
20ð?1Þ þ 2ae?ix~ sWð1Þ
20ð0Þ þ 2Wð1Þ
11ð0Þ þ aeix~ sWð1Þ
?o
20ð0Þ
??
11ð0Þ þ Wð2Þ
20ð0Þ
?
:
ð3:10Þ
_W ¼ _ xt? _ zq ?_? z? q ¼
AW ? 2Ref? q?ð0Þf0qðhÞg;
AW ? 2Ref? q?ð0Þf0qð0Þg þ f0;
h 2 ½?1;0Þ;
h ¼ 0;
¼
defAW þ Hðz;? z;hÞ;
ð3:11Þ
Hðz;? z;hÞ ¼ H20ðhÞz2
Expanding the above series and comparing the corresponding coefficients, we obtain
2þ H11ðhÞz? z þ H02ðhÞ? z2
2þ ???:
ð3:12Þ
ðA ? 2x~ sÞW20ðhÞ ¼ ?H20ðhÞ;AW11ðhÞ ¼ ?H11ðhÞ;...:
From (3.11), we know that for h 2 [?1,0),
Hðz;? z;hÞ ¼ ?? q?ð0Þf0qðhÞ ? q?ð0Þf0? qðhÞ ¼ ?gqðhÞ ? ? g? qðhÞ:
Comparing the coefficients with (3.12) gives that
ð3:13Þ
ð3:14Þ
H20ðhÞ ¼ ?g20qðhÞ ? ? g02? qðhÞ;
ð3:15Þ
and
H11ðhÞ ¼ ?g11qðhÞ ? ? g11? qðhÞ:
From (3.13), (3.15) and the definition of A , it follows that
ð3:16Þ
_W20ðhÞ ¼ 2ix~ sW20ðhÞ þ g20qðhÞ þ ? g02? qðhÞ:
Notice that qðhÞ ¼ ð1;a;bÞTeihxksk, hence
W20ðhÞ ¼ig20
x~ sqð0Þeihx~ sþi? g02
3x~ s? qð0Þe?ihx~ sþ E1e2ihx~ s;
ð3:17Þ
where E1¼ ðEð1Þ
Similarly, from (3.13) and (3.16), we can obtain
W11ðhÞ ¼ ?ig11
where E2¼ ðEð1Þ
In what follows, we shall seek appropriate E1and E2. From the definition of A and (3.13), we obtain
Z0
and
Z0
1;Eð2Þ
1Þ 2 R2is a constant vector.
x~ sqð0Þeihx~ sþi? g11
x~ s? qð0Þe?ihx~ sþ E2;
ð3:18Þ
2;Eð2Þ
2Þ 2 R2is also a constant vector.
?1
dgðhÞW20ðhÞ ¼ 2ix~ sW20ð0Þ ? H20ð0Þð3:19Þ
?1
dgðhÞW11ðhÞ ¼ ?H11ð0Þ;
ð3:20Þ
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where g(h) = g(0,h). From (3.11) and (3.12), we have
H20ð0Þ ¼ ?g20qð0Þ ? ? g02? qð0Þ ? 2~ srx?
a1a2e?ix~ sþ a2a
0
??
ð3:21Þ
and
H11ð0Þ ¼ ?g11qð0Þ ? ? g11? qð0Þ ? 2~ srx?
a1jaj2Refeix~ sg þ a2Refag
0
!
:
ð3:22Þ
Substituting (3.17) and (3.21) into (3.19) and noticing that
Z0
and
Z0
we obtain
Z0
which leads to
?
ix~ sI ?
?1
eihx~ sdgðhÞ
??
qð0Þ ¼ 0
?ix~ sI ?
?1
e?ihx~ sdgðhÞ
??
? qð0Þ ¼ 0;
2ix~ sI ?
?1
e2ihx~ sdgðhÞ
??
E1¼ ?2rx?
a1a2eix~ sþ a2a
0
??
;
2ix þ ra1x?e?2ix~ s
?r
It follows that
ra2x?
2ix þ r
?
E1¼ ?2rx?
a1a2eix~ sþ a2a
0
??
:
Eð1Þ
1 ¼?2rx?ðr þ 2ixÞða1a2eix~ sþ a2aÞ
A
and
Eð2Þ
1 ¼?2rx?rða1a2eix~ sþ a2aÞ
A
;
where
A ¼
2ix þ ra1x?e?2ix~ s
?r
ra2x?
2ix þ r
????????:
Similarly, substituting (3.18) and (3.22) into (3.20), we can obtain
ra1x?
?r
ra2x?
r
??
E2¼ 2rx?
a1jaj2Refeix~ sg þ a2Refag
0
!
:
and hence,
Eð1Þ
2 ¼ Eð2Þ
2 ¼
2 a1jaj2Refeix~ sg þ a2Refag
a1þ a2
??
:
Therefore, we can determine W20(0) and W11(0) from (3.17) and (3.18). Furthermore, we can determine g21.
Therefore, each gijin (3.9) is determined by the parameters and delay in (3.10). Thus, we can compute the fol
lowing values:
Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757
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c1ð0Þ ¼
i
2~ sx
g11g20? 2jg11j2?jg02j2
3
!
þg21
2;
l2¼ ?Refc1ð0Þg
Refk0ð~ sÞg;
b2¼ 2Refc1ð0Þg;
T2¼ ?Imfc1ð0Þg þ l2Imfk0ð~ sÞg
~ sx
;
ð3:23Þ
which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value ~ s,
i.e., l2determines the directions of the Hopf bifurcation: if l2> 0(l2< 0), then the Hopf bifurcation is super
critical (subcritical) and the bifurcating periodic solutions exist for s > ~ sðs < ~ sÞ; b2determines the stability of
the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if b2< 0 (b2> 0); and
T2determines the period of the bifurcating periodic solutions: the period increase (decrease) if T2> 0 (T2< 0).
These, together with (2.8), (2.9), Lemma 2.1 and Theorem 2.2, imply the following.
Theorem 3.1. (i) Suppose that D > 0,a2> a1and 0 < r <
s ¼ sþ
inverse conclusion holds for s ¼ s?
supercritical) if Re{c1(0)} < 0 (respectively, Re{c1(0)} > 0). On the center manifold, the Hopf bifurcating periodic
solution at E*for s ¼ s?
holds, then system (2.1) undergoes a Hopf bifurcation at sþ
(respectively, subcritical) if Re{c1(0)} < 0 (respectively, Re{c1(0)} > 0). On the center manifold, the Hopf
bifurcating periodic solution are stable(unstable) if b2< 0 (b2> 0).
rða2þ
ffiffiffiffiffiffiffiffiffi
a2
2þa2
1
p
a2þa1
Þ
. Then the Hopf bifurcations of (2.1) at
jare supercritical (respectively, subcritical) if Re{c1(0)} < 0 (respectively, Re{c1(0)} > 0). However, the
j, namely, the Hopf bifurcations of (2.1) at s ¼ s?
jare subcritical (respectively,
jare stable(unstable) if b2< 0 (b2> 0). (ii) If either a2< a1or a2= a1and 0 < r <
j, and the Hopf bifurcations are supercritical
ffiffi
2
2r
p
þ1
A numerical example. From the above algorithm, we know that if the values of r,r,ai(i = 1,2) and s are
given, then we can compute the values of l2and b2determine the stability and direction of periodic solutions
bifurcating from the positive equilibrium at the critical point s?
results by considering the system:
j. In the sequel, we illustrate the validity of the
_ xðtÞ ¼ xðtÞ½1 ? 2xðt ? sÞ ? yðtÞ?;
_ yðtÞ ¼ xðtÞ ? yðtÞ;
which has a positive equilibrium E?¼ ð1
2jp
0:797728, the positive equilibrium E*is stable when s < sþ
bifurcation at sþ
?
ð3:24Þ
3;1
3Þ. It follows from Section 2 and Theorem 2.2 that sþ
0(see Figs. 1–3), and system (3.24) undergoes a Hopf
j. Further, from the above process, we can determine the stability and direction of periodic
j¼:2:77303þ
Fig. 1. Behavior of x in system (3.24) with s = 2.3 and the initial value x0= 0.8.
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Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757
Page 11
solutions bifurcating from the positive equilibrium at the critical point sþ
we can compute c1(0) G ?0.399728 ? 0.00886001I. It then follows that l2> 0 and b2< 0. Therefore, the
bifurcating periodic solutions exists at least for the value s slightly large than sþ
odic orbits are orbitally asymptotically stable, as depicted in Figs. 4–6.
j. For instance, when s ¼ sþ
0¼:2:77303,
0and the corresponding peri
Fig. 2. Behavior of y in system (3.24) with s = 2.3 and the initial value y0= 0.8.
Fig. 3. Phase portrait of system (3.24) with s = 2.3, (x0,y0) = (0.8,0.8). The positive equilibrium (x*,y*) of system (3.24) is stable for
s < sþ
0.
Fig. 4. Periodic oscillation of x in system (3.24) with s = 2.8 and the initial value x0= 0.8.
Y. Song, Y. Peng / Applied Mathematics and Computation 181 (2006) 1745–1757
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4. Conclusions
The Logistic equation with one or two discrete delays has been widely studied by authors. For the delayed
Logistic equation (1.1), it is well known that (i) the positive equilibrium x ? k is asymptotically stable for any
s 2 ½0;p
x ? k at s ¼1
the distributed delay (i.e., a1> a2), then Eq. (1.4) almost has the same dynamics as the Logistic equation (1.1)
with a discrete delay. However, Theorem 2.2 shows that the dynamics of the Logistic equation with discrete
and distributed delays are more rich and interesting. For example, the phenomena of stability switches occur if
D > 0,a2> a1and 0 < r <
a2þa1
2rÞ, and unstable for s >p
rfp
2r, and (ii) Eq. (1.1) undergoes a Hopf bifurcation from the positive equilibrium
2þ 2jpg. From Theorem 2.2, we obtain that if the effect of the discrete delay is large than that of
rða2þ
ffiffiffiffiffiffiffiffiffi
a2
2þa2
1
p
Þ
.
References
[1] G.S. Jones, The existence of periodic solution of f
[2] S.N. Chow, J. MalletParet, Integral averaging and bifurcation, J. Differential Equations 26 (1977) 112–159.
[3] A. Stech, Hopf bifurcation calculations for functional differential equations, J. Math. Anal. Appl. 109 (1985) 472–491.
[4] I. Gyo ¨ri, G. Ladas, Oscillation theory of delay differential equations. With applications, Oxford Mathematical Monographs, Oxford
Science Publications. The Clarendon Press, Oxford University Press, New York, 1991.
[5] J. Hale, S.M. Lunel, Introduction to Functional Differential Equations, Springer, New York, 1993.
[6] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.
0(x) = ?af(x ? 1)[1 + f(x)], J. Math. Anal. Appl. 5 (1962) 435–450.
Fig. 6. Phase portrait of system (3.24) with s = 2.8. Here initial values are taken as (0.4,0.4) and (0.8,0.8), respectively. Clearly, this
numerical simulation shows that the bifurcating periodic solution is asymptotically stable.
Fig. 5. Periodic oscillation of y in system (3.24) with s = 2.8 and the initial value y0= 0.8.
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[7] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and its Applications,
vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1992.
[8] J.M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, Springer, Heidelberg, 1977.
[9] B.D. Hassard, N.D. Kazarinoff, Y.H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge,
1981.
[10] K. Cooke, Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl. 86 (1982) 592–627.
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